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<h1 class="heading"><a href="MATH-2023-OPDE.html"><span class="title">MATH 2023: Ordinary and Partial Differential Equations</span></a></h1>
<p class="byline">Xiaoyi Chen and Wei Zhang</p>
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<a href="ch_first.html" data-scroll="ch_first" class="internal"><span class="codenumber">1</span> <span class="title">Introduction</span></a><ul>
<li><a href="sec_1-intro.html" data-scroll="sec_1-intro" class="internal">Classification of Differential Equations</a></li>
<li><a href="sec_2-intro.html" data-scroll="sec_2-intro" class="internal">Linear and Nonlinear Equation</a></li>
<li><a href="sec_3-intro.html" data-scroll="sec_3-intro" class="internal">Geometrical Aspect</a></li>
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<a href="ch_second.html" data-scroll="ch_second" class="internal"><span class="codenumber">2</span> <span class="title">First Order Ordinary Differential Equations</span></a><ul>
<li><a href="sec2_1.html" data-scroll="sec2_1" class="internal">Linear Equations</a></li>
<li><a href="sec2_2.html" data-scroll="sec2_2" class="internal">Further Discussion of Linear Equations (For reading only)</a></li>
<li><a href="sec2_3.html" data-scroll="sec2_3" class="internal">Separable Equations</a></li>
<li><a href="sec2_4.html" data-scroll="sec2_4" class="internal">Difference Between Linear and Nonlinear Equations</a></li>
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<li><a href="sec2_6.html" data-scroll="sec2_6" class="internal">Exact Equations and Integrating Factors</a></li>
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<a href="ch_third.html" data-scroll="ch_third" class="internal"><span class="codenumber">3</span> <span class="title">third Order Linear Equations</span></a><ul>
<li><a href="sec3_1.html" data-scroll="sec3_1" class="internal">Homogeneous equations with constant coefficient</a></li>
<li><a href="sec3_2.html" data-scroll="sec3_2" class="internal">Fundamental Solutions of Linear Homogeneous Equations</a></li>
<li><a href="sec3_3.html" data-scroll="sec3_3" class="internal">Linear Independence and Wronskian</a></li>
<li><a href="sec3_4.html" data-scroll="sec3_4" class="internal">Complex roots of the characteristic equations</a></li>
<li><a href="sec3_5.html" data-scroll="sec3_5" class="internal">Repeated Roots: Reduction of Order</a></li>
<li><a href="sec3_6.html" data-scroll="sec3_6" class="internal">Non-homogeneous Equations and Method of Undetermined Coefficients</a></li>
<li><a href="sec3_7.html" data-scroll="sec3_7" class="internal">Variation of Parameters</a></li>
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<li><a href="sec4_1.html" data-scroll="sec4_1" class="internal">General Theory of the <span class="process-math">\(n\)</span>-th Order Linear Equations</a></li>
<li><a href="sec4_2.html" data-scroll="sec4_2" class="internal">Homogeneous Equations with Constant Coefficients</a></li>
<li><a href="sec4_3.html" data-scroll="sec4_3" class="internal">The Method of Undetermined Coefficients</a></li>
<li><a href="sec4_4.html" data-scroll="sec4_4" class="internal">The Method of Variation of Parameters</a></li>
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<a href="ch_five.html" data-scroll="ch_five" class="internal"><span class="codenumber">5</span> <span class="title">Series Solutions of Second Order Linear Equations</span></a><ul>
<li><a href="sec5_1.html" data-scroll="sec5_1" class="internal">Brief Review on Power Series</a></li>
<li><a href="sec5_2.html" data-scroll="sec5_2" class="internal">Introduction</a></li>
<li><a href="sec5_3.html" data-scroll="sec5_3" class="internal">Series Solutions Near an Ordinary Point</a></li>
<li><a href="sec5_4.html" data-scroll="sec5_4" class="internal">Euler’s Equation</a></li>
<li><a href="sec5_5.html" data-scroll="sec5_5" class="internal">Series Solution near a Regular Singular Point</a></li>
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<a href="ch_six.html" data-scroll="ch_six" class="internal"><span class="codenumber">6</span> <span class="title">System of First Order Linear Equations</span></a><ul>
<li><a href="sec6_1.html" data-scroll="sec6_1" class="internal">Introduction <span class="process-math">\(\&amp;\)</span> Basic Theory</a></li>
<li><a href="sec6_2.html" data-scroll="sec6_2" class="internal">Homogeneous System with Constant Coefficients</a></li>
<li><a href="sec6_3.html" data-scroll="sec6_3" class="internal">Complex Eigenvalues</a></li>
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<li><a href="sec6_5.html" data-scroll="sec6_5" class="internal">Fundamental Matrices</a></li>
<li><a href="sec6_6.html" data-scroll="sec6_6" class="active">Non-homogeneous linear systems</a></li>
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<a href="ch_seven.html" data-scroll="ch_seven" class="internal"><span class="codenumber">7</span> <span class="title">Partial Differential Equations</span></a><ul>
<li><a href="sec7_1.html" data-scroll="sec7_1" class="internal">Two-Point Boundary Value Problems</a></li>
<li><a href="sec7_2.html" data-scroll="sec7_2" class="internal">Eigenvalue Problems</a></li>
<li><a href="sec7_3.html" data-scroll="sec7_3" class="internal">Fourier Series</a></li>
<li><a href="sec7_4.html" data-scroll="sec7_4" class="internal">The Fourier Convergence Theorem</a></li>
<li><a href="sec7_5.html" data-scroll="sec7_5" class="internal">Even and Odd Functions</a></li>
<li><a href="sec7_6.html" data-scroll="sec7_6" class="internal">Introduction to Partial Differential Equations</a></li>
<li><a href="sec7_7.html" data-scroll="sec7_7" class="internal">1D Heat Equation; Solutions by Separation of Variable and Fourier Series</a></li>
<li><a href="sec7_8.html" data-scroll="sec7_8" class="internal">Other Heat Conduction Problems</a></li>
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<a href="ch_eight.html" data-scroll="ch_eight" class="internal"><span class="codenumber">8</span> <span class="title">Laplace transform</span></a><ul>
<li><a href="sec8_1.html" data-scroll="sec8_1" class="internal">What are Laplace Transforms, and Why?</a></li>
<li><a href="sec8_2.html" data-scroll="sec8_2" class="internal">Finding Laplace Transforms</a></li>
<li><a href="sec8_3.html" data-scroll="sec8_3" class="internal">Finding inverse transforms using partial fractions</a></li>
<li><a href="sec8_4.html" data-scroll="sec8_4" class="internal">Solving ODEs and ODE Systems</a></li>
<li><a href="sec8_5.html" data-scroll="sec8_5" class="internal">Step input and Impulse problems</a></li>
<li><a href="sec8_6.html" data-scroll="sec8_6" class="internal">Laplace transform for PDE (heat equation)</a></li>
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<li class="link"><a href="solutions-1.html" data-scroll="solutions-1" class="internal"><span class="codenumber">A</span> <span class="title">Selected Hints</span></a></li>
<li class="link"><a href="solutions-2.html" data-scroll="solutions-2" class="internal"><span class="codenumber">B</span> <span class="title">Selected Solutions</span></a></li>
<li class="link"><a href="appendix-1.html" data-scroll="appendix-1" class="internal"><span class="codenumber">C</span> <span class="title">List of Symbols</span></a></li>
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<main class="main"><div id="content" class="pretext-content"><section class="section" id="sec6_6"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">6.6</span> <span class="title">Non-homogeneous linear systems</span>
</h2>
<p id="p-283">Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="" id="eq8_4">
\begin{equation}
{\bf x}^{\prime}={\bf P}(t)\,{\bf x}+{\bf g}(t),\quad \alpha \leq t \leq \beta,\tag{6.6.1}
\end{equation}
</div>
<p class="continuation">where <span class="process-math">\({\bf P}(t)\)</span> and <span class="process-math">\({\bf g}(t)\)</span> are assumed to be continuous on <span class="process-math">\(\alpha \leq t \leq \beta\text{.}\)</span></p>
<p id="p-284">Suppose that <span class="process-math">\({\bf x}_1\)</span> is a particular solution to (<a href="" class="xref" data-knowl="./knowl/eq8_4.html" title="Equation 6.6.1">(6.6.1)</a>) and the general solution for</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_4.html ./knowl/eq8_4.html" id="eq8_5">
\begin{equation}
{\bf x}^{\prime}={\bf P}(t)\,{\bf x},\quad \alpha \leq t \leq \beta,\tag{6.6.2}
\end{equation}
</div>
<p class="continuation">is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_4.html ./knowl/eq8_4.html">
\begin{equation*}
C_1 {\bf x}^{(1)}+C_2 {\bf x}^{(2)}+\cdots+C_n {\bf x}^{(n)}.
\end{equation*}
</div>
<p class="continuation">Then</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_4.html ./knowl/eq8_4.html">
\begin{equation*}
{\bf x}=C_1 {\bf x}^{(1)}+C_2 {\bf x}^{(2)}+\cdots+C_n {\bf x}^{(n)}+{\bf x}_1
\end{equation*}
</div>
<p class="continuation">is the general solution to (<a href="" class="xref" data-knowl="./knowl/eq8_4.html" title="Equation 6.6.1">(6.6.1)</a>).</p>
<p id="p-285"><dfn class="terminology">Variation of Parameters</dfn>The general solution to (<a href="" class="xref" data-knowl="./knowl/eq8_5.html" title="Equation 6.6.2">(6.6.2)</a>) may be written as</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_5.html ./knowl/eq8_6.html">
\begin{equation*}
{\bf x}={\bm \Psi}(t)\, {\bf c},
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\({\bm \Psi}(t)\)</span> is the fundamental matrix. To find a particular solution, we let</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_5.html ./knowl/eq8_6.html" id="eq8_6">
\begin{equation}
{\bf x}_1={\bm \Psi}(t)\, {\bf u}(t).\tag{6.6.3}
\end{equation}
</div>
<p class="continuation">We want to determine <span class="process-math">\({\bf u}(t)\)</span> such that (<a href="" class="xref" data-knowl="./knowl/eq8_6.html" title="Equation 6.6.3">(6.6.3)</a>) is a solution.</p>
<p id="p-286">Substitute (<a href="" class="xref" data-knowl="./knowl/eq8_6.html" title="Equation 6.6.3">(6.6.3)</a>) into (<a href="" class="xref" data-knowl="./knowl/eq8_4.html" title="Equation 6.6.1">(6.6.1)</a>):</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_6.html ./knowl/eq8_4.html ./knowl/eq8_7.html ./knowl/eq8_8.html ./knowl/eq8_1.html" id="eq8_7">
\begin{equation}
{\bm \Psi}^{\prime}(t)\, {\bf u}(t)+{\bm \Psi}(t)\, {\bf u}^{\prime}(t)={\bf P}(t) {\bm \Psi}(t) {\bf u}(t)+{\bf g}(t),\tag{6.6.4}
\end{equation}
</div>
<p class="continuation">where</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_6.html ./knowl/eq8_4.html ./knowl/eq8_7.html ./knowl/eq8_8.html ./knowl/eq8_1.html" id="eq8_8">
\begin{equation}
{\bm \Psi}^{\prime}(t)=\left[{\bf x}^{\prime (1)}, {\bf x}^{\prime (2)},\cdots, {\bf x}^{\prime (n)}\right]
=\left[{\bf P}(t)\, {\bf x}^{(1)}, {\bf P}(t)\, {\bf x}^{(2)}, \cdots, {\bf P}(t)\, {\bf x}^{(n)}\right]
={\bf P}(t) \left[{\bf x}^{(1)}, {\bf x}^{(2)}, \cdots, {\bf x}^{(n)}\right]
={\bf P}(t) \, {\bm \Psi}(t).\tag{6.6.5}
\end{equation}
</div>
<p class="continuation">Based on (<a href="" class="xref" data-knowl="./knowl/eq8_7.html" title="Equation 6.6.4">(6.6.4)</a>) and (<a href="" class="xref" data-knowl="./knowl/eq8_8.html" title="Equation 6.6.5">(6.6.5)</a>)</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_6.html ./knowl/eq8_4.html ./knowl/eq8_7.html ./knowl/eq8_8.html ./knowl/eq8_1.html">
\begin{equation*}
{\bm \Psi}(t)\, {\bf u}^{\prime}(t)={\bf g}(t).
\end{equation*}
</div>
<p class="continuation">Multiplying <span class="process-math">\({\bm \Psi}^{-1}(t)\text{,}\)</span> we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_6.html ./knowl/eq8_4.html ./knowl/eq8_7.html ./knowl/eq8_8.html ./knowl/eq8_1.html">
\begin{equation*}
{\bf u}^{\prime}(t)={\bm \Psi}^{-1}(t)\,{\bf g}(t) \to 
{\bf u}(t)=\int {\bm \Psi}^{-1}(t)\,{\bf g}(t) \mathrm{d} t+k_1,
\end{equation*}
</div>
<p class="continuation">where we may choose <span class="process-math">\(k_1=0\text{.}\)</span> The general solution for (<a href="" class="xref" data-knowl="./knowl/eq8_1.html" title="Equation 6.5.1">(6.5.1)</a>) is then</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_6.html ./knowl/eq8_4.html ./knowl/eq8_7.html ./knowl/eq8_8.html ./knowl/eq8_1.html">
\begin{equation*}
{\bf x}={\bm \Psi}(t)\, {\bf c}+{\bm \Psi}(t)\, \int {\bm \Psi}^{-1}(t)\,{\bf g}(t) \mathrm{d} t.
\end{equation*}
</div>
<p id="p-287"><dfn class="terminology">Example 1</dfn> Find the general solution of</p>
<div class="displaymath process-math" data-contains-math-knowls="" id="eq8_10">
\begin{equation}
{\bf x}^{\prime}={\bf A} \, {\bf x}+{\bf g}(t)=
\left(
\begin{array}{cc}
2 &amp; -5\\
1 &amp; -2
\end{array}
\right)\, {\bf x}+\left(
\begin{array}{c}
-5 \cos t\\
\sin t
\end{array}
\right).\tag{6.6.6}
\end{equation}
</div>
<p id="p-288"><dfn class="terminology">Solution</dfn> Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
({\bf A} -r{\bf I} ) \vec{\xi}={\bf 0}.
\end{equation*}
</div>
<p class="continuation">We have</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
|{\bf A} -r{\bf I} |=0 \to
\left|
\begin{array}{cc}
2-r &amp; -5\\
1 &amp; -2-r
\end{array}
\right|=0 \to 
r_1=i,\quad r_2=-i. \quad (\lambda=0, \mu=1)
\end{equation*}
</div>
<p id="p-289">For <span class="process-math">\(r_1=i\text{,}\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\left(
\begin{array}{cc}
2-i &amp; -5\\
1 &amp; -2-i
\end{array}
\right)
\left(
\begin{array}{c}
\xi^{(1)}_1\\
\xi^{(1)}_2
\end{array}
\right)={\bf 0},
\end{equation*}
</div>
<p class="continuation">which gives</p>
<div class="displaymath process-math" data-contains-math-knowls="" id="eq8_9">
\begin{equation}
\begin{array}{c}
(2-i) \xi^{(1)}_1-5 \xi^{(1)}_2=0\\
\xi^{(1)}_1-(2+i)  \xi^{(1)}_2=0.
\end{array}\tag{6.6.7}
\end{equation}
</div>
<p class="continuation">Note <span class="process-math">\((\ref{eq8_9})_1 \times (2+i)\)</span> leads to <span class="process-math">\(\xi^{(1)}_1-(2+i)  \xi^{(1)}_2=0\)</span> which is exactly <span class="process-math">\((\ref{eq8_9})_2\text{.}\)</span> From <span class="process-math">\((\ref{eq8_9})_2\text{,}\)</span> we let <span class="process-math">\(\xi^{(1)}_1=5\text{,}\)</span> then <span class="process-math">\(\xi^{(1)}_2=2-i\text{.}\)</span> So</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\vec{\xi}^{(1)}=\left(
\begin{array}{c}
5\\
2-i
\end{array}
\right)=\left(
\begin{array}{c}
5\\
2
\end{array}
\right)+i \left(
\begin{array}{c}
0\\
-1
\end{array}
\right)={\bf a}+i {\bf b}.
\end{equation*}
</div>
<p id="p-290">The two solutions are</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
&amp;{\bf x}^{(1)}=e^{\lambda t}({\bf a} \cos \mu t-{\bf b} \sin \mu t)=\left(
\begin{array}{c}
5\\
2
\end{array}
\right) \cos t-\left(
\begin{array}{c}
0\\
-1
\end{array}
\right) \sin t=\left(
\begin{array}{c}
5 \cos t\\
2 \cos t+\sin t
\end{array}
\right),\\
&amp;{\bf x}^{(2)}=e^{\lambda t}({\bf b} \cos \mu t+{\bf a} \sin \mu t)=\left(
\begin{array}{c}
0\\
-1
\end{array}
\right) \cos +\left(
\begin{array}{c}
5\\
2
\end{array}
\right) \sin t=\left(
\begin{array}{c}
5 \sin t\\
-\cos t+2 \sin t
\end{array}
\right).
\end{aligned}
\end{equation*}
</div>
<p class="continuation">The fundamental matrix is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bm \Psi}(t)=\left(
\begin{array}{cc}
5 \cos t &amp; 5 \sin t\\
2 \cos t+\sin t &amp;-\cos t+2 \sin t
\end{array}
\right).
\end{equation*}
</div>
<p id="p-291">For a particular solution, we let</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_10.html">
\begin{equation*}
{\bf x}^{(p)}={\bm \Psi}(t)\,{\bf u}(t).
\end{equation*}
</div>
<p class="continuation">Substituting it into (<a href="" class="xref" data-knowl="./knowl/eq8_10.html" title="Equation 6.6.6">(6.6.6)</a>), we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_10.html">
\begin{equation*}
{\bm \Psi}(t) \, {\bf u}^{\prime}(t)={\bf g}(t) \to
\left(
\begin{array}{cc}
5 \cos t &amp; 5 \sin t\\
2 \cos t+\sin t &amp;-\cos t+2 \sin t
\end{array}
\right) \left(
\begin{array}{c}
u_1^{\prime}\\
u_2^{\prime}
\end{array}
\right)=\left(
\begin{array}{c}
-5 \cos t\\
\sin t
\end{array}
\right)
\end{equation*}
</div>
<p class="continuation">which further gives</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_10.html">
\begin{equation*}
u_1^{\prime}=-\cos 2 t+\sin 2 t,\quad
u_2^{\prime}=-2 \cos t (\cos t+\sin t).
\end{equation*}
</div>
<p class="continuation">Integration gives one set of solutions to <span class="process-math">\(u_1, u_2\)</span> as</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_10.html">
\begin{equation*}
u_1=-\frac{1}{2} \cos 2t-\frac{1}{2} \sin 2 t,\quad
u_2=-t +\cos ^2 t-\frac{1}{2} \sin 2t.
\end{equation*}
</div>
<p class="continuation">The general solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq8_10.html">
\begin{equation*}
\begin{aligned}
{\bf x}&amp;={\bm \Psi}(t)\,{\bf c}+{\bm \Psi}(t)\,{\bf u}(t)\\
&amp;= \left(-\frac{1}{2} \cos 2t-\frac{1}{2} \sin 2 t+C_1\right) \left(
\begin{array}{c}
5 \cos t\\
2 \cos t+\sin t
\end{array}
\right)+\left(-t +\cos ^2 t-\frac{1}{2} \sin 2t+C_2 \right) \left(
\begin{array}{c}
5 \sin t\\
-\cos t+2 \sin t
\end{array}
\right).
\end{aligned}
\end{equation*}
</div>
<p id="p-292"><dfn class="terminology">Diagonalization</dfn> Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf x}^{\prime}={\bf A}\,{\bf x}+{\bf g}(t),\quad \alpha \leq t \leq \beta,
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\({\bf A}\)</span> is a constant coefficient matrix. Introduce <span class="process-math">\({\bf x}={\bf T}\, \vec{y}\text{,}\)</span> where <span class="process-math">\({\bf T}=(\vec{\xi}^{(1)}, \vec{\xi}^{(2)}, \cdots, \vec{\xi}^{(n)})\)</span> and <span class="process-math">\(\vec{\xi}^{(1)}, \vec{\xi}^{(2)}, \cdots, \vec{\xi}^{(n)}\)</span> are eigenvectors corresponding to eigenvalues <span class="process-math">\(r_1, r_2, \cdots, r_n\text{.}\)</span> Then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf T} \,\vec{y}^{\prime}={\bf A}\,{\bf T}\, \vec{y}+{\bf g}(t) \to
\vec{y}^{\prime}={\bf T}^{-1}\,{\bf A}\,{\bf T}\, \vec{y}+{\bf T}^{-1}\,{\bf g}(t) =
\left(
\begin{array}{cccc}
r_1 &amp; 0 &amp; \cdots &amp; 0\\
0&amp;r_2&amp;\cdots&amp;0\\
\vdots &amp; \vdots &amp; \ddots &amp; \vdots\\
0 &amp; 0&amp; \vdots &amp; r_n
\end{array}
\right)\vec{y}+\left(
\begin{array}{c}
h_1(t)\\
h_2(t)\\
\vdots\\
h_n(t)
\end{array}
\right).
\end{equation*}
</div>
<p class="continuation">In component form</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{array}{c}
y_1^{\prime}=r_1 y_1+h_1(t)\\
y_2^{\prime}=r_2 y_2+h_2(t)\\
\vdots\\
y_n^{\prime}=r_n y_n+h_n(t)\\
\end{array}
\end{equation*}
</div>
<p class="continuation">which are decoupled equations. Using the method of integrating factor, we have solutions</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{array}{c}
y_1=C_1 e^{r_1 t}+e^{r_1 t} \int e^{-r_1 t} h_1(t) \mathrm{d} t,\\
y_2=C_2 e^{r_2 t}+e^{r_2 t} \int e^{-r_2 t} h_2(t) \mathrm{d} t,\\
\vdots\\
y_n=C_n e^{r_n t}+e^{r_n t} \int e^{-r_n t} h_n(t) \mathrm{d} t.\\
\end{array}
\end{equation*}
</div>
<p class="continuation">The solution is then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf x}={\bf T}\,\vec{y}.
\end{equation*}
</div>
<p id="p-293"><dfn class="terminology">Example 2</dfn> Find the general solution of</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf x}^{\prime}={\bf A}\,{\bf x}+{\bf g}(t)
=
\left(
\begin{array}{cc}
1 &amp; 1\\
4 &amp; -2
\end{array}
\right) {\bf x}+\left(
\begin{array}{c}
e^{-2t}\\
-2 e^t
\end{array}
\right).
\end{equation*}
</div>
<p id="p-294"><dfn class="terminology">Solution</dfn> Two eigenvalues for <span class="process-math">\({\bf A}\)</span> are <span class="process-math">\(r_1=-3, r_2=2\text{.}\)</span> Two corresponding eigenvectors are</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\vec{\xi}^{(1)}=\left(
\begin{array}{c}
1\\
-4
\end{array}
\right),\quad
\vec{\xi}^{(2)}=\left(
\begin{array}{c}
1\\
1
\end{array}
\right).
\end{equation*}
</div>
<p class="continuation">Thus,</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf T}=\left(
\begin{array}{cc}
1 &amp; 1\\
-4 &amp; 1
\end{array}
\right).
\end{equation*}
</div>
<p id="p-295">Let <span class="process-math">\({\bf x}={\bf T}\,\vec{y}\text{,}\)</span> then the equation becomes</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\vec{y}^{\prime}={\bf T}^{-1} \, {\bf A}\, {\bf T}\,\vec{y}+{\bf T}^{-1}  \, {\bf g}(t),
\end{equation*}
</div>
<p class="continuation">where</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf T}^{-1}=\frac{1}{5}\left(
\begin{array}{cc}
1 &amp; -1\\
4 &amp; 1
\end{array}
\right),\quad
{\bf T}^{-1}  \, {\bf g}(t)=\frac{1}{5}\left(
\begin{array}{c}
e^{-2t}+2 e^t\\
4 e^{-2t}-2e^t
\end{array}
\right).
\end{equation*}
</div>
<p class="continuation">In component form, we have</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{array}{c}
y_1^{\prime}=-3y_1+\frac{1}{5} (e^{-2t}+2 e^t),\\
y_2^{\prime}=2y_2+\frac{1}{5} (4 e^{-2t}-2 e^t),
\end{array}
\end{equation*}
</div>
<p class="continuation">which lead to solutions</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{array}{c}
y_1=C_1 e^{-3t}+\frac{1}{5}(e^{-2t}+e^t/2 )\\
y_2=C_2 e^{2t}+\frac{1}{5}(2 e^t-e^{-2t}).
\end{array}
\end{equation*}
</div>
<p class="continuation">Then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf x}={\bf T}\,\vec{y}=C_1 \left(
\begin{array}{c}
1\\
-4
\end{array}
\right) e^{-3t}+C_2 \left(
\begin{array}{c}
1\\
1
\end{array}
\right) e^{2t}+ \left(
\begin{array}{c}
0\\
-1
\end{array}
\right) e^{-2t}+ \left(
\begin{array}{c}
1/2\\
0
\end{array}
\right) e^{t}.
\end{equation*}
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